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Pure Core
1. Complex Numbers
1.3 Argand Diagrams
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An Argand diagram is a graphical representation of complex numbers on a two-dimensional
plane
Complex numbers in the form
a
+
a +
a
+
b
i
bi
bi
are plotted with
a
a
a
on the real axis and
b
b
b
on the imaginary axis.
How would the complex number
3
+
3 +
3
+
4
i
4i
4
i
be plotted on an Argand diagram?
(3, 4)
Match the axis with its representation:
Horizontal Axis ↔️ Real axis
Vertical Axis ↔️ Imaginary axis
Steps to plot a complex number on an Argand diagram:
1️⃣ Identify the real part
2️⃣ Identify the imaginary part
3️⃣ Plot the point (a, b) on the plane
The real axis on an Argand diagram represents the imaginary part of a complex number.
False
Where is the complex number
2
+
2 +
2
+
3
i
3i
3
i
plotted on an Argand diagram?
(2, 3)
What is an Argand Diagram used for?
Representing complex numbers
In an Argand Diagram, the real part of a complex number is plotted on the
horizontal
The imaginary axis in an Argand Diagram corresponds to the y-axis in a
Cartesian Plane
.
What is the vertical axis in an Argand Diagram called?
Imaginary axis
Match the components of an Argand Diagram with their counterparts in a Cartesian Plane:
Horizontal Axis ↔️ x-axis
Vertical Axis ↔️ y-axis
Where is the complex number
3
+
3 +
3
+
4
i
4i
4
i
plotted on an Argand Diagram?
(3, 4)
To plot a complex number on an Argand Diagram, you must identify its
real
and imaginary parts.
Steps to plot a complex number on an Argand Diagram:
1️⃣ Identify the real and imaginary parts.
2️⃣ Locate the point on the diagram.
3️⃣ Mark the point to represent the complex number.
The real axis in an Argand Diagram is
horizontal
.
What are the two axes in an Argand Diagram called?
Real and imaginary axes
In an Argand Diagram, complex numbers are plotted using their
real
and imaginary parts.
What are the axes of an Argand Diagram called?
Real and imaginary axes
The imaginary axis in an Argand Diagram is
vertical
Match the diagram type with its axis names:
Argand Diagram ↔️ Real axis ||| Imaginary axis
Cartesian Plane ↔️ x-axis ||| y-axis
The complex number
3
+
3 +
3
+
4
i
4i
4
i
is plotted as (3, 4) on an Argand Diagram.
How do you plot a complex number
a
+
a +
a
+
b
i
bi
bi
on an Argand Diagram?
Locate point (a, b)
The modulus of a complex number z = a + bi</latex> is its distance from the
origin
The formula for the modulus of
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is
∣
z
∣
=
|z| =
∣
z
∣
=
\sqrt{a^{2} +
b^{2}}
.
What does the modulus of a complex number represent graphically?
Distance from the origin
The modulus of a complex number is always a
positive
real number.
The argument of a complex number
z
=
z =
z
=
a
+
a +
a
+
b
i
bi
bi
is the angle
θ
\theta
θ
calculated as \theta = \tan^{ - 1}\left(\frac{b}{a}\right)</latex>.origin
What is the argument of
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
in terms of
tan
−
1
\tan^{ - 1}
tan
−
1
?
tan
−
1
(
4
3
)
\tan^{ - 1}\left(\frac{4}{3}\right)
tan
−
1
(
3
4
)
The argument of a
complex number
is measured from the positive real axis.
What is the argument of
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
?
tan
−
1
(
4
3
)
\tan^{ - 1}\left(\frac{4}{3}\right)
tan
−
1
(
3
4
)
What is the argument of a complex number defined as?
The angle
θ
\theta
θ
between the positive real axis and the line connecting the origin to
z
z
z
on an Argand Diagram
The argument of a complex number z = a + bi</latex> is calculated as
θ
=
\theta =
θ
=
tan
−
1
(
b
a
)
\tan^{ - 1}\left(\frac{b}{a}\right)
tan
−
1
(
a
b
)
On which diagram is the argument of a complex number represented graphically?
Argand Diagram
For
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
, the argument is
θ
=
\theta =
θ
=
tan
−
1
(
4
3
)
\tan^{ - 1}\left(\frac{4}{3}\right)
tan
−
1
(
3
4
)
, which is the angle in radians
What does the calculation of the argument of a complex number depend on?
The quadrant of
z
z
z
In the first quadrant, if
a
>
0
a > 0
a
>
0
and
b
>
0
b > 0
b
>
0
, then
θ
=
\theta =
θ
=
tan
−
1
(
b
a
)
\tan^{ - 1}\left(\frac{b}{a}\right)
tan
−
1
(
a
b
)
What is the formula for
θ
\theta
θ
in the second quadrant when
a
<
0
a < 0
a
<
0
and
b
>
0
b > 0
b
>
0
?
θ
=
\theta =
θ
=
π
+
\pi +
π
+
tan
−
1
(
b
a
)
\tan^{ - 1}\left(\frac{b}{a}\right)
tan
−
1
(
a
b
)
What is the argument of a complex number defined as?
Angle between positive real axis and z
The argument for
z
=
z =
z
=
3
+
3 +
3
+
4
i
4i
4
i
is calculated as
tan
−
1
(
4
3
)
\tan^{ - 1}\left(\frac{4}{3}\right)
tan
−
1
(
3
4
)
, which simplifies to \theta
See all 60 cards
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